Paperfolding
puzzles are puzzles that are solved by folding
paper. They are not just origami versions of
puzzles which were originally created in some
other way. In order to qualify as a paperfolding
puzzle paperfolding must be integral to the
solving of the puzzle not just to its
construction. So, for instance, a set of Tangram
pieces folded from paper is not a paperfolding
puzzle. Nor is a set of Soma Cubes made out of
folded paper modules. Similarly a puzzle that is
set by folding paper but is solved by the use of
mathematics does not qualify as a paperfolding
puzzle. Paperfolding may also be used to create,
or set, the puzzle, but that is not integral to
the definition. It follows that paperfolding puzzles
are not inherently difficult to solve. Their
solutions are easily accessible through the
intelligent and persistent application of trial
and error. Play around with the possibilities
long enough and a solution will emerge. It is
worth bearing in mind, however, that many
paperfolding puzzles have multiple solutions, and
that the first solution you come across will not
necessarily be the simplest or the best.
As a result
of this multiplicity of solutions I have not been
able to include solutions to every part of every
puzzle, or, where I know that multiple solutions
exist, to include all the solutions I am aware
of. Space simply does not permit this. It is also
quite conceivable that I have not found the best
solutions. They may still be waiting for you to
discover them.
Puzzles
have rules, or conditions. In puzzles set using
more robust materials, such as wood, the
conditions are often entirely physical. Will the
pieces go together or not? In paperfolding
puzzles, however, the conditions tend to be
rather more subtle. I have taken great care to
try to state the conditions for each puzzle
clearly, and you should take equal care to make
sure you understand them before you start looking
for the solution. This should not mean, however,
that you should not feel free to think about
unusual possibilities.
All the
puzzles in this book start from squares but they
require two different types of paper. Most
require irogami, which is paper that is white one
side and a single plain colour the other. The
others require homogeneous paper, which is the
same colour on both sides. Irogami is commonly
sold as ‘origami paper’ in packs of
precut squares. Homogeneous paper usually comes
in rectangles. The best source is photocopy
paper. Pages 133 and 134 show you how to cut
rectangles down to squares.
Some of the
puzzles, and many more of the solutions, rely on
the division of the width of the paper into
three, four, five or more equal parts, or on
grids that are created by dividing both the width
and the height of the paper in this way.
Sometimes estimating these divisions by eye will
work perfectly well. It is, however, often
preferable, and at times essential, to be able to
do this division accurately. For this reason,
three supplements showing you how to fold a 4x4
grid, and how to divide into thirds and fifths
are included at the back of this book. You can
adapt the methods shown there to allow you to
divide a square into any other number of equal
parts.
Paperfolding
puzzles do not easily fall into neat compartments
within some overall scheme. There are several
different ways to divide them into categories,
all of which are useful and illuminating. One
such distinction can be drawn between unfolded
sheet, grid and apparatus puzzles. As the name
suggests, unfolded sheet puzzles start from one
or more unfolded sheets of paper, the challenge
being to fold them freestyle, and perhaps also
assemble them, until the solution is achieved.
Grid puzzles start from a sheet of paper that has
already been folded into a grid of creases. Only
the creases in the grid may be used to achieve
the solution. Apparatus puzzles start from paper
that has been cut and glued together into some
kind of simple apparatus which is then folded, or
manipulated, until the solution is achieved.
Paperfolding
puzzles can also, using established origami
language, be characterised as pure or,
presumably, impure origami puzzles, a pure
origami puzzle being one that can be set and
solved without the use of cuts, decoration or
adhesives. Most of the puzzles in this book are
pure origami puzzles in this sense.
Alternatively,
and perhaps more usefully, paperfolding puzzles
can be divided into eight broad categories
related to the object of the puzzle. These are
shape forming puzzles, pattern forming puzzles,
layering puzzles, tabletop puzzles, fold and cut
puzzles, transformation puzzles, discovery
puzzles and assembly puzzles. These categories
are not, however, always mutually exclusive.
Shape
forming puzzles: In shape forming
puzzles the object is to fold the paper to match
the shape of a flat motif. Merlin’s Mat,
Nimue’s Mat, Foursquare and Juxtaposition
are all examples of this type of puzzle.
Pattern
forming puzzles: In pattern forming
puzzles the object is to fold a piece of irogami
(paper which is white one side and coloured the
other) to match a specified target pattern. For
convenience, all the pattern forming puzzles in
this book start from an unfolded square, but
there is no compelling reason why this should be
the case. In many cases the target pattern is
also square, the exceptions being Equality, where
the finished pattern can be of any shape, and
Challenge 1 of Striptease, where the challenge is
to create the desired pattern within the largest
possible rectangle. All the target patterns in
this book are flat but there is no reason why the
challenge could not be, for instance, to create
the pattern on all the surfaces of a cube using
identical modules.
The aim of
pattern forming puzzles is generally not only to
achieve the target pattern but also to achieve it
in the smallest possible number of folds. This
requires some explanation. Technically a fold is
a change of direction in the paper. When you
flatten a fold you get a crease. So folding is a
process and a crease is the result of this
process. Counting folds and creases is not always
the same. If you lay two sheets of paper together
and fold them in half you could argue that you
have made one fold but two creases. You can also
make a fold without making a crease. Sometimes
you have to begin to solve a pattern making
puzzle by making one or more construction folds
(folds that are not used in a solution but help
locate other folds that are). For instance if you
want to fold one corner of a square into the
centre you need to know where the centre is. You
can find this centre by creasing in both
diagonals. The centre is where they cross.
Construction creases do not count towards the
total number of creases required to solve the
puzzle.
The
majority of the puzzles in this book are pattern
forming puzzles, and of these the majority are
singlesided patterns (i.e. the pattern is
created on just one side of the paper).
Doublesided pattern puzzles (where the pattern
is created on both sides of the paper) are not
necessarily harder to solve but they are more
complicated to understand since there are several
variants to consider. Examples of all four types
of doublesided pattern are given on page 31. You
may need to go and lie down after you have
studied them.
Layering
puzzles: The object of layering
puzzles is to fold a sheet of paper so that the
corners, or other clearly identifiable parts,
such as certain squares within a larger grid, lie
on top of each other in a given order. Sequential
is a simple puzzle of this kind.
Tabletop
puzzles: A tabletop puzzle is a folding
puzzle where some part of one or other surface of
the paper must remain in contact with the top of
a table (or a similar hard surface) while each of
the folds (or unfolds) that lead to a solution is
being made. There are two puzzles of this kind,
Turnover and Inside Out, in this book.
Fold
and Cut puzzles: The challenge of a Fold
and Cut puzzle is to fold a sheet of paper in
such a way that the target shape can be cut from
the paper using just a single straight cut. Three
such puzzles, Square Cut, Cross Cut and
Staircase, are included in this book. It is worth
noting that, despite the name, Fold and Cut
puzzles are pure origami puzzles. The solution is
achieved just by folding the paper. The cut
simply confirms that the solution has been found.
Transformation
puzzles: The object of transformation
puzzles is to change one state of a puzzle into
another. There is a sense in which every
paperfolding puzzle is a transformation puzzle,
but I usually reserve the term for apparatus
puzzles such as Centrepiece where the possibility
for such a transformation seems unlikely.
Transformation puzzles can be set up so that the
aim is to find a route that avoids one obvious
state. I have not included any of this type of
puzzle in this book.
Discovery
puzzles: A discovery puzzle is a kind of
transformation puzzle which includes a state that
is difficult to access or can only be found as
the result of some lateral thinking. The Windmill
Flexagon is a simple example of this kind of
puzzle.
Assembly
puzzles: The object of assembly puzzles
is to find a way to put folded paper modules
together to create a target shape. Since they are
not solved by folding paper, but rather by
assembling prefolded or partly prefolded
modules, there is a sense in which these puzzles
do not fit into the core concept of this book.
However, the assembly aspect of modular origami
is so much an integral part of modern day origami
design that I have chosen to include two
examples, Quartetra and 2 into 1, despite this
consideration.
*
It is
perhaps worth briefly considering what it is that
goes to make a good paperfolding puzzle. I think
it is a combination of three things, simplicity,
misdirection and surprise. A good origami puzzle
should have a solution that is not immediately
obvious, but is equally not beyond the reach of
paperfolders who apply their fingers and
imaginations to solving the challenge in a
focused way. Ideally, there should be that 'Ah,
yes ...' moment when the solution suddenly
appears, and seems thereafter to have been
obvious from the beginning. Solutions to simple
puzzles often have this quality. The more complex
the puzzle, the more likely it is that it can be
solved logically rather than through a sudden
flash of insight.
The first
edition of this book was published in 1998.
Except for Black and White, TTime and the
Flexatube, this second edition contains all the
puzzles and solutions found in the original
edition plus many additions and amendments.
Unless otherwise stated all the puzzles and
puzzle solutions in this book were originated by
the author.
In a book
of this kind it is always tempting to move
straight from the puzzle to the solution. Please
try to resist this temptation. I have had the
pleasure of creating these puzzles, and working
out their solutions, over many years and I would
like you to share this same enjoyment. If you go
straight from the puzzle to the solution not only
will you miss out on the pleasure of finding at
least one solution for yourself but also be
unable to evaluate how easy or difficult the
challenges are, or to appreciate the combination
of mental and physical skills that are needed to
solve them.
David
Mitchell
